General Transport Equations
Dimensional analysis helps us to understand much about a
system, especially what is important and what is not;
however, if we are to be able to calculate things when
forming a design, we need to formulate equations.
Fluid mechanics are based on conservation or transport
equations. The things being transported by a moving fluid
are mass, momentum, and energy.
Exact equations can be derived by examining a small
volume of fluid. This will be done here for a Cartesian
system, but the same equations can be derived for an
arbitrary coordinate system.
Resources
• White, chapter 4, sections 4.1 to 4.6
• Alexandrou, chapter 5
• Virtual wind tunnel: http://raphael.mit.edu/Java/
• More links are available on the website
Use of general equations
• Not all systems fit into neat categories.
• Some systems are complicated.
• Many different factors that can change completely the
methods used to model a system and subsequent results.
• These factors include: unsteadiness (changes with time);
non-uniformity (changes in space); compressibility
(changes in density); stresses („viscosity‟); No. dimensions.
Example: turbofan engine
Non-uniform and everything changing: how can we cope
with it?
Source: http://en.wikipedia.org/wiki/Image:Turbofan_operation.png
Computational Fluid Dynamics
(CFD)
Meshing
Further information about CFD available via web site.
Gradients
x
f(x)
changes with x
x
f
Over small distances, the curve is
nearly straight, the gradient is
approximately constant and
How does CFD calculate flows?
• Divide the flow into very small elements.
• Over a small element, gradients are approximately equal to the
linear gradient over the small element.
• For each element, solve the simplified equations for the fluid
flow through them.
• Equations express conservation of mass, momentum, and
(sometimes) energy.
• Solution for one element feeds into the solution for the
neighbours.
• At the edge of the fluid flow, boundary conditions are
required.
• If the flow is unsteady (changes with time), initial conditions
are required.
Gradients in more than one
dimension
Gradient= tan
Gradient zero
Gradient between zero and tan
i.e. Gradient depends on direction and
therefore is a vector
Gradient operatorGradient is then represented by a vector operator , pronounced „grad‟. It is called an operator because it cannot appear on its own, it
always needs to operate on a variable e.g.
In a Cartesian coordinate system (different forms exist for other
coordinate systems- see handout.
Or, for a vector u=u(u, v, w),
There is also the Laplacian operator
One-dimensional mass transport
Summary
• Control volume small enough for all changes
over it to be approximately linear.
• Control volume small enough for the product of
changes to be negligible (δ20).
• Compare mass flow rate in with mass flow out.
• Compare result with rate of change of mass in the
control volume.
(Remember “rate” means gradient with time)
Three-dimensional mass transport
Fluxes
Mass is just one of the quantities that can be transported
by a fluid.
Volume flow rate of fluid through control volume in x-
direction= speed area = u y z
If is a quantity per unit volume transported by the fluid,
then its rate of transport in the x-direction= u y z.
The flux of (flow rate per unit area) in the x-direction=
u.
Therefore generally, the flux of a quantity per unit
volume transported by a fluid is
Different fluxes
e ueueEnergy
u uuuuMomentum
uuMass
Flux in
three-
dimensions
Flux in x-
direction
Quantity
per unit
volume
Conservation of Momentum
Newton‟s Second Law of Motion
Rate of change of momentum=sum of forces
Momentum flux per unit volume= uu
Rate of change of momentum per unit volume in the
control volume
Derive this from first principles as for the
conservation of mass.
Total derivative
The total, material, or substantial derivative is
given by
Necessary for field variables i.e. those that change with
time and position = (x,t).
Rate of
change of
temperature
= / t
Rate of change
of temperature
=
(dx/dt)( / x)x
Total derivatives
g
Act on all the fluid
within the control
volume
Body forces
Force per unit volume
Static forces
Act on all the surface
of the control volume.
Act whether the fluid
is moving or not.
Far face hidden.
p
p
p
p+( p/ x) x
p+( p/ y) y
xz
y
Direct stresses and shear stresses
Direct stresses act in a direction perpendicular to the
surface of a control volume. In a fluid they can be
static or dynamic.
Shear stresses act in a direction parallel to the surface
of a control volume. It acts as if to shear a control
volume. Only a dynamic force in a fluid i.e. the fluid
has to be moving.
xx
xy
xz
zx
zy
zz
yx
yy
yz
Act on the surface
of the control
volume only when
the fluid is moving
Note can be both direct
stresses and shear
stresses.
y
z
x
yx+ yx
yy+ yy
yz+ yz
xx+ xx
xy+ xy
xz+ xz
Far face hidden
Dynamic forces
Dynamic surface forces in the x-
direction
xx
yx
zx
x
y
First subscript refers to the face on which the stress is acting.
The second subscript refers to the direction of the stress.
Stress Tensor
The stress tensor is the combination of the static and dynamic
surfaces forces. In many engineering situations, this is the difficult
thing to specify e.g. turbulent flows, multiphase flows, non-
Newtonian fluids.
zzyzxz
zyyyxy
zxyxxx
p
p
p
σ
Equation for the conservation of
momentum
Advective
component
Static
surface
forces
Dynamic
surface
forces
Body
forces
Navier-Stokes relations
Assumes fluid is Newtonian i.e. stress strain and isotropic.
Practical momentum equations
• The Navier-Stokes equations are exact, but
unsolvable; therefore, they need to be simplified
for practical engineering calculations.
• Different systems require different
simplifications.
• As a rule of thumb, you want to have the simplest
equations possible to describe a particular
system.
Simplifications (1)• Viscosity ( )= constant. Not true if significant changes in
temperature.
• Density ( ) =constant i.e.the flow is incompressible. Not true if there are large temperature variations or the flow is fast moving.
Mass
Momentum
These are often the equations solved by CFD packages
and misnamed as the Navier-Stokes equations.
Simplifications (2)
• =0 i.e.fluid is inviscid. Means that forces owing
to viscosity are negligible, not viscosity is negligible.
gu
pDt
DEuler‟s equation
Simplifications (3)
• u/ t=0 i.e. flow is steady.
• Number of dimensions.
One-dimensional, steady, incompressible,
inviscid flow results in Bernoulli equation.
Show this to yourself.
Example
A confectionary manufacturer is designing a machine for the icing of biscuits, as shown in figure Q1. Starting from the general momentum equation describe the factors that have to be taken into account to obtain a working momentum equation, and why these are relevant to this particular situation. All relevant factors should be taken into account, but you should pay particular attention to the stress tensor.
Icing
Biscuit
Figure Q1
Complications
• Multiphase flows
• Non-Newtonian fluids
• Turbulence
In particular, the difficulty is stating and solving the
stress tensor.